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%I #16 Jun 09 2019 18:43:23
%S 1,2,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,27,
%T 28,29,31,32,33,34,35,36,37,38,39,41,43,44,45,46,47,48,49,50,51,52,53,
%U 55,57,58,59,61,62,63,64,65,67,68,69,71,73,74,75,76,77,79,80,81,82,83,84
%N Infinitary deficient numbers: integers for which A126168(n) < n, or equivalently for which A049417(n) < 2n.
%C For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~1.144n. It follows that the density of the infinitary deficient numbers is 1/1.144, which is about 0.874.
%H Amiram Eldar, <a href="/A129657/b129657.txt">Table of n, a(n) for n = 1..10000</a>
%H Graeme L. Cohen, <a href="http://dx.doi.org/10.1090/S0025-5718-1990-0993927-5">On an Integer's Infinitary Divisors</a>, Mathematics of Computation, Vol. 54, No. 189, (1990), pp. 395-411.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InfinitaryDivisor.html">Infinitary Divisor</a>.
%e The sixth integer that exceeds its proper infinitary divisor sum is 7. Hence a(6)=7.
%t ExponentList[n_Integer,factors_List]:={#,IntegerExponent[n,# ]}&/@factors;InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f,g}, BitOr[f,g]==g][ #,Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #,factors]&/@d]],_?(And@@#&),{1}]] ]] ] Null;properinfinitarydivisorsum[k_]:=Plus@@InfinitaryDivisors[k]-k;InfinitaryDeficientNumberQ[k_]:=If[properinfinitarydivisorsum[k]<k,True,False];Select[Range[100],InfinitaryDeficientNumberQ[ # ] &] (* end of program *)
%t fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[100], isigma[#] < 2 # &] (* _Amiram Eldar_, Jun 09 2019 *)
%Y Cf. A126168, A049417, A127666, A129656, A007357.
%K easy,nonn
%O 1,2
%A _Ant King_, Apr 29 2007
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